Force fields

Question 1

Explain the idea of a atom-centered force field.

Answer 1

Instead of considering the wave functions of the electrons to describe the interaction between atoms are the electrons only considered as an electric continuum defined by an electric permittivity eps –> Born-Oppenheimer approximation.
The force field only depends on the position of the atom.

Question 2

Explain the bond term in the force field.

Answer 2

Between atoms in a molecule an equilibrium bond length is defined, when the bond length is displaced from this equilibrium a force acts to bring it back in place, like a spring.
The potential is a sum of all the bonds in the molecule.
V(r1,r2,..,rn) = sum(1/2*k_bond (bi-bi,0)^2
where bi,0 is the equilibrium bond length for bond i and k_bond the force constant.

Question 3

Explain the angle term in the force field.

Answer 3

It is similar to the bond term, it is also modelled as a spring, with an equilibrium angle theta_i,0.
V(theta_1,…,theta_n) = sum(1/2*k_angle (theta_i -theta_i,0)^2

Question 4

Explain the torsion term in the force field.

Answer 4

The torsion is like a fourier series and it controls the rotation around the bonds, so it can form different dihedrals. Like cis/trans conformation.
The potential:
V = sum all torsions( sum over n(ktao_n,i (1+cos(n_itao_i-delta_i))))
more terms can be added in the series to change the behavior of the potential.
eg. for ethane there are three H with 120° in between them all, this gives a periodic potential with min at 60°, 180° and 300°.
But when changing one of the hydrogen atoms for an Cl for example the heights of the min and max change, which is realised by adding terms in the series.

Question 5

Explain how to construct the artificial potential controlling the chirality.

Answer 5

To control that switching chirality gives another molecule, planes are defined from their normals between the atoms of the molecule.
An angle is defined as alpha = arccos(n_ABCn_ADB), for example and the potential is then:
V_chiral = 1/2K (alpha-alpha_0)^2
when the normal of the plane changes, the potential change.

Question 6

Explain where the attractive and repulsive part of the Lennard-Joneese potential comes form.

Answer 6

The repulsive part is due to Pauli exclusion, electrons are fermions and cannot be in the same state. This means that two atoms cannot be too close.
Prop. to (sigma/r)^12

The attractive part comes from the van der Waals interactions, which comes form dipole-dipole interactions.
There is a min in the LJ-pot. at some distance which is the most favourable inter-particle distance for the particles.
Prop. to (sigma/r)^6

Question 7

Explain how to construct the potential due to the charges in the molecule.

Answer 7

Considering the electrons as a continuum, one can integrate over the volume of the molecule to find the electrostatic pot. from the electron cloud.
The electrostatic pot. from one small volume element dV on a test charge outside the electron cloud is:
dE(r_test) = rho_e(r)q_edV/(4pieps abs(r-r_test))
integrating this over the whole volume gives the contribution from all the electrons, and then the contribution from the nucleus at R_i, with atom number Z_i, has to be added:
sum(Z_i*q_e/abs(R_i-r_test)

But then the charge of these two contributions are fitted to a param. and then one only needs to sum over all the nuclei in the molecule.

Question 8

How can the parameters for the force field be obtained from experiments?

Answer 8

From high resolution crystals equilibrium bond lengths can be obtained.
Vibrational bond stretching can be obtained from IR spectroscopy.
Or by QM calculations can energy barriers for bond stretching, bending etc. be obtained.